I need to compute integral lines on a plane of a vector field which itself is a numerical integral over a 3rd dimension. Being fairly new to Mathematica I'm not sure of the best way to go about this.

An obvious approach I'm trying is to start at a point (x0,y0), find the field vector V0, project a short distance along it to a new point (x1,y1), and find the field vector V1 there. [This can be iterated to improve accuracy by refining (x1,y1) to be projected along (V0+V1)/2.] Using a For loop this procedure can be repeated to move along the generated integral curve, and another enveloping For loop can move the initial starting point to generate a family of integral curves.

But is there a better way to do this? Does Mathematica already have a tool for this operation? Thanks.